Variational Approximation of a Second Order Free Discontinuity Problem in Computer Vision

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(1)c 2001 Society for Industrial and Applied Mathematics . Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. SIAM J. MATH. ANAL. Vol. 32, No. 6, pp. 1171–1197. VARIATIONAL APPROXIMATION OF A SECOND ORDER FREE DISCONTINUITY PROBLEM IN COMPUTER VISION∗ LUIGI AMBROSIO† , LORIS FAINA‡ , AND RICCARDO MARCH§ Abstract. We consider a functional, proposed by Blake and Zisserman for computer vision problems, which depends on free discontinuities, free gradient discontinuities, and second order derivatives. We show how this functional can be approximated by elliptic functionals defined on Sobolev spaces. The approximation takes place in a variational sense, the De Giorgi Γ-convergence, and extends to this second order model an approximation of the Mumford–Shah functional obtained by Ambrosio and Tortorelli. For the purpose of illustration an algorithm based on the Γ-convergent approximation is applied to the problem of computing depth from stereo images and some numerical examples are presented. Key words. theory and algorithms for computer vision, variational problems, Γ-convergence, functions of bounded variation AMS subject classifications. 46E30, 49J45 PII. S0036141000368326. 1. Introduction. In recent years, variational principles with a free discontinuity set have been introduced to solve reconstruction problems in computer vision theory (see, for instance, [4, 26, 31]). The variational approach to the image segmentation problem proposed by Mumford and Shah [27] consists of minimizing the functional (1.1). E(u, K) =. Ω\K. . |∇u|2 + µ|u − g|2 dx + αHn−1 (K ∩ Ω) ,. where Ω ⊂ Rn is a bounded open set, Hn−1 is the Hausdorff (n − 1)-dimensional measure, g ∈ L∞ (Ω), and α, µ > 0 are fixed positive parameters. The functional has to be minimized over all closed sets K ⊂ Ω and all u ∈ C 1 (Ω\K). In the case n = 2 the function g represents the image to be segmented. By minimizing the functional one tries to detect the discontinuities of g due to the edges of the objects in the image, and to cancel the discontinuities due to noise and small irregularities. The set K contains the jump points of u and represents the edges of the objects. The functional penalizes large sets K, and outside K the function u is required to be close to g and C 1 . The Mumford and Shah variational principle can be extended to several reconstruction problems of computer vision [25]: stereo reconstruction [32], computation of optical flow [28], shape from shading [33]. Variational problems involving functionals of this form are usually called free discontinuity problems, after a terminology introduced by De Giorgi [18]. ∗ Received by the editors February 22, 2000; accepted for publication (in revised form) October 3, 2000; published electronically February 28, 2001. http://www.siam.org/journals/sima/32-6/36832.html † Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy (luigi@ambrosio.sns.it). The work of the first author was partially supported by Progetto Nazionale MURST “Equazioni Differenziali e Calcolo delle Variazioni” 9701226040. ‡ Dipartimento di Matematica, Via L. Vanvitelli 1, 06123 Perugia, Italy (faina@unipg.it). § Istituto per le Applicazioni del Calcolo, CNR, Viale del Policlinico 137, 00161 Roma, Italy (march@iac.rm.cnr.it). The work of the third author was partially supported by Progetto MURST 5% “Rete Multimediale, LdR: Applicazione ai Beni Culturali.”. 1171.

(2) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1172. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. The Mumford and Shah model has some drawbacks: it is unable to reconstruct crease discontinuities and yields the over-segmentation of steep gradients (the socalled ramp effect). To overcome these defects of the first order model, Blake and Zisserman [9] introduced a second order functional which can be written in the form 2 2 F (u, K0 , K1 ) = |∇ u| + Φ(x, u) dx Ω\(K0 ∪K1 ) n−1. + αH. (1.2). (K0 ∩ Ω) + βHn−1 ((K1 \K0 ) ∩ Ω) ,. with α, β > 0 positive parameters. The functional has to be minimized over the unknown sets K0 , K1 , with K0 ∪ K1 closed and u ∈ C 2 (Ω\(K0 ∪ K1 )) approximately continuous on Ω\K0 . If Φ(x, u) = µ|u − g|2 and n = 2, the functional (1.2) is just that one introduced in [9] (the thin plate surface under tension). In the second order model, K0 represents the set of jump points for u, and K1 \K0 is the set of crease points. Since the reconstruction of crease discontinuities is particularly relevant in those computer vision problems which require the reconstruction of visible surfaces from two-dimensional images, we have then introduced in (1.2) the function Φ(x, u). A suitable choice of this function will allow us to apply this variational method to computer vision problems as, for instance, the computation of depth from pairs of stereo images (see [25]). If the conditions (see [9]) (1.3). β ≤ α ≤ 2β. are satisfied, the existence of minimizers for the functional F (u, K0 , K1 ) has been proved, in the case n = 2 and Φ(x, u) = µ|u − g|2 , by Carriero, Leaci, and Tomarelli [13] (notice that (1.3) are necessary and sufficient for the lower semicontinuity of F with respect to the L1 convergence). The proof is based on a weak formulation of the problem by setting 2 2 F (u) = |∇ u| + Φ(x, u) dx + αHn−1 (Su ) + βHn−1 (S∇u \Su ) , (1.4) Ω. where ∇u denotes an approximate differential, Su is the discontinuity set of u in an approximate sense, and S∇u is the discontinuity set of ∇u. In [12] the existence of minimizers for the functional F over the space u : Ω → R : u ∈ L2 (Ω), u ∈ GSBV (Ω), ∇u ∈ [GSBV (Ω)]n , (1.5) has been proved in any space dimension n, GSBV (Ω) being the space of generalized special functions of bounded variation introduced in [17]. A regularity theorem in [13] then shows that, for n = 2, any weak minimizer actually provides a minimizing triplet (u, K0 , K1 ) of F by taking a suitable representative of the function and the closure of Su and S∇u . Ambrosio and Tortorelli [5, 6] approximated the Mumford and Shah functional (1.1) by a family of elliptic functionals defined on Sobolev spaces. The approximation takes place in a variational sense, the De Giorgi Γ-convergence. The approximating elliptic functionals proposed in [6] are defined by E (u, s) = (s2 + λ )|∇u|2 dx + µ (1.6) |u − g|2 dx + αG (s) , Ω. Ω.

(3) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1173. where the approximation takes place as → 0+ , λ → 0+ , and (s − 1)2 2 (1.7) dx . |∇s| + G (s) = 4 Ω The variable s ∈ [0, 1] is related to the set of jumps K. The minimizing s are near to 0 in a neighborhood of the set K, and far from the neighborhood they are close to 1. The neighborhood shrinks as → 0. The Ambrosio and Tortorelli approximation can be used to find an effective algorithm for computing the minimizers of E [25, 29, 30]. The approximation has been applied to several computer vision problems in [28, 32, 33], and further improvements have been proposed and experimented in [34]. In the present paper we consider the following family of functionals: F (u, s, σ) = (σ 2 + κ )|∇2 u|2 dx + Φ(x, u) dx + (α − β)G (s) Ω Ω + βG (σ) + ξ (s2 + ζ )|∇u|γ dx (1.8) Ω. for suitable infinitesimals κ , ξ , ζ , and γ ≥ 2. A slight variant of these functionals has been proposed by Bellettini and Coscia [7] in the case n = 1 and in that case the Γ-convergence of F to F has been proved (see also the discussion in the beginning of section 6). We extend their Γ-convergence result in the following way: we prove the lower inequality of Γ-convergence in any space dimension n, and we prove the upper inequality when u is bounded and |∇u| ∈ Lγ (Ω), under a very mild regularity assumption on the sets Su and S∇u , which is fulfilled in computer vision applications. In the particular case when α = β and n = 2, we obtain a full Γ-convergence theorem. The extension of the Ambrosio and Tortorelli approximation to the second order problem presents several difficulties. The lower inequality cannot be obtained by means of the slicing technique and consequent reduction to a one-dimensional problem used in [5, 6]. Such a reduction yields the operator norm of the Hessian matrix in the Γ-limit instead of the euclidean norm. The second derivatives are then estimated by adapting a global technique proposed by Ambrosio in [3] and relying on a compactness theorem in the space (1.5) due to Carriero, Leaci, and Tomarelli [12]. Conversely, the jump part of the functional is estimated by using a slicing argument, taking into account that the space GSBV is a vector space under a suitable energy condition (Proposition 4.3). The major difficulty in the proof of the upper inequality consists in obtaining a suitable estimate on |∇u|γ dx from the finiteness of (1.4). Such an estimate would permit us to adapt the constructive part of Ambrosio and Tortorelli’s proof [6] to the second order problem. In the case α = β, n = γ = 2, an estimate which yields a full Γ-convergence result is obtained by means of a suitable interpolation inequality in W 2,2 (Proposition 4.6). If α = β, we obtain only a partial result, proving the upper inequality under some mild regularity assumptions on u. The discretization of the functional (1.2) is not straightforward and it is difficult to apply gradient descent with respect to the unknown sets K0 and K1 . Conversely, the Γ-convergent approximation yields a sequence of functionals (1.8) which are numerically much more tractable, so that discretization and gradient descent may be applied in a straightforward way. In particular, a simple discretization method, commonly used for computer vision problems [35], may be applied to the functionals (1.8). We then apply the Γ-convergence result to the problem of computation of depth from stereo images, and we present some computer experiments on synthetic images to illustrate the feasibility of the approximation..

(4) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1174. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. 2. Notations and preliminary results. Let Ω ⊂ Rn be a bounded open set. We denote by B(Ω) the σ-algebra of all the Borel subsets of Ω; for any C ∈ B(Ω) we denote by meas(C) the Lebesgue n-dimensional measure of C and by Hn−1 (C) the Hausdorff (n − 1)-dimensional measure of C. We denote by Bρ (x) the open ball {y ∈ Rn : |y − x| < ρ}. We denote by Mn×n the space of n × n matrices endowed with the euclidean norm. We introduce the following notations: s ∧ t = min{s, t}, s ∨ t = max{s, t} for every s, t ∈ R; given two vectors a, b, we set a, b = a · b = i ai bi and (a ⊗ b)ij = ai bj . For any Borel function u : Ω → R we define the approximate upper and lower limits u+ (x), u− (x) by. meas({y ∈ Bρ (x) : u(y) > t}) = 0 , u+ (x) = inf t ∈ [−∞, +∞] : lim ρn ρ→0+. meas({y ∈ Bρ (x) : u(y) < t}) − =0 . u (x) = sup t ∈ [−∞, +∞] : lim+ ρn ρ→0 The set Su = {x ∈ Ω : u− (x) < u+ (x)} is the discontinuity set of u in an approximate sense and it is negligible with respect to Lebesgue measure (see [20, section 2.9.13]). Suppose z = u+ (x) = u− (x) ∈ R; we say that ∇u(x) ∈ Rn is the approximate differential of u at x if v + (x) = 0, where v(y) =. |u(y) − z − ∇u(x), y − x| |y − x|. ∀y ∈ Ω \ {x} .. If u is differentiable at x, then ∇u(x) is the classical gradient. In the one-dimensional case we shall use the notation u in place of ∇u. An important property of the approximate differential is the fact that (2.1) ∇u(x) = 0. almost everywhere (a.e.) on. {y ∈ Ω : u(y) = c}. ∀c ∈ R .. We denote by BV (Ω) the space of functions of bounded variation in Ω, i.e., the functions u ∈ L1 (Ω) such that the distributional derivative of u is representable by means of a vector measure Du = (D1 u, . . . , Dn u) with finite total variation. We denote by |Du| the measure total variation of Du. If u ∈ BV (Ω), then ∇u exists a.e. in Ω and coincides a.e. with the Radon–Nikodym derivative of Du with respect to the Lebesgue measure [11]. Moreover, the set Su is countably (n − 1)-rectifiable, n−1 i.e., representable as a disjoint union ∪∞ (N ) = 0 and Ki are i=1 Ki ∪ N , where H 1 n compact sets, each contained in a C hypersurface Γi ⊂ R [16]. Let E ⊂ B(Ω); we define. . P (E, Ω) = sup div φ dx : φ ∈ C01 (Ω; Rn ), |φ| ≤ 1 . E. We say that E is a set of finite perimeter in Ω if P (E, Ω) < +∞. By Riesz’s theorem (see [21]), E is a set of finite perimeter if and only if 1E ∈ BV (Ω), and P (E, Ω) = |D1E |(Ω). The following Fleming–Rishel coarea formula (see [21]) establishes an important connection between BV functions and sets of finite perimeter: +∞ (2.2) P ({x ∈ Ω : u(x) > t}, Ω) dt. |Du|(Ω) = −∞.

(5) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1175. We say that u ∈ BV (Ω) belongs to the space of special functions of bounded variation SBV (Ω) if |∇u| dx + |u+ − u− | dHn−1 . |Du|(Ω) = Ω. Su. Functions like the Cantor–Vitali function, whose derivative is concentrated on Cantor’s middle third set, are then excluded by SBV (Ω) (see [1, 17]). Given a Borel function u : Ω → R we say that u ∈ GSBV (Ω) if (see [2, 17]) −N ∨ u ∧ N ∈ SBVloc (Ω). (2.3). ∀N ∈ N.. The jump set of u is given by Su =. ∞

(6). S−N ∨u∧N .. N =1. Furthermore, if u ∈ GSBV (Ω), then Su is countably (n − 1)-rectifiable, ∇u exists a.e. in Ω and is given by (see [2]) ∇u = ∇(−N ∨ u ∧ N ). a.e. on {x ∈ Ω : |u| ≤ N }. ∀N ∈ N.. We also set GSBV 2 (Ω) = {u ∈ GSBV (Ω) : ∇u ∈ [GSBV (Ω)]n } . Given u ∈ GSBV 2 (Ω), we use the notation ∇2i,j u = ∇j (∇i u) and, in the onedimensional case, u. = (u ) . Moreover we set S∇u =. n

(7). S∇ i u .. i=1. The following compactness result has been proved by Carriero, Leaci, and Tomarelli in [12]. Theorem 2.1. Let (uh ) ⊂ GSBV 2 (Ω) be a sequence such that Hn−1 (Suh ∪ S∇uh ) , |∇2 uh |2 dx uh L2 , Ω. are uniformly bounded in h. Then there exist a subsequence (uhk ) and u ∈ GSBV 2 (Ω)∩ L2 (Ω) such that, as k → +∞, uhk → u strongly in L1 (Ω), ∇uhk → ∇u a.e. in Ω, 2 2 ∇ uhk - ∇ u weakly in L2 (Ω; Mn×n ) . Finally, we recall the following lemma (see [10]). Lemma 2.2. Let µ : B(Ω) → [0, +∞] be a σ-finite measure, and let (fi ) ⊂ L1 (Ω) be a sequence of nonnegative functions. Then, sup fi (x) dµ(x) Ω i∈N. = sup. k i=1. Ai. fi (x) dµ(x) : Ai ⊂ Ω open and mutually disjoint, k ∈ N ..

(8) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1176. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. We now recall the definition and some properties of Γ-convergence (see [15]). Let X be a metric space and let f : X → [0, +∞] be a family of functions indexed by > 0. We say that f Γ-converge as → 0+ to f : X → [0, +∞] if the following two conditions (2.4). ∀x → x. lim inf f (x ) ≥ f (x) +. ∃x → x. lim sup f (x ) ≤ f (x). →0. and (2.5). →0+. are fulfilled for every x ∈ X. The Γ-limit, if it exists, is unique and lower semicontinuous. The Γ-convergence is stable under continuous perturbations, that is, f + g Γ-converge to f + g if f Γ-converge to f and g is continuous. The most important property of Γ-convergence is the following: if (x ) is asymptotically minimizing, i.e., (2.6) lim+ f (x ) − inf f = 0, X. →0. and if xh converge to x for some sequence h → 0, then x minimizes f . 3. Statement of main results. Let Φ(x, u) = µ|u − g|2 , µ > 0, and 0 < β ≤ α ≤ 2β. For every u ∈ GSBV 2 (Ω) ∩ L2 (Ω) and every g ∈ L∞ (Ω), we write (1.4) as 2 2 |∇ u| + µ|u − g|2 dx + (α − β)Hn−1 (Su ) + βHn−1 (Su ∪ S∇u ). F (u) = Ω. In [12], using Theorem 2.1 and a suitable lower semicontinuity theorem in GSBV 2 (Ω), Carriero, Leaci, and Tomarelli proved that the problem (P) min F (u) : u ∈ GSBV 2 (Ω) ∩ L2 (Ω) has at least one solution. For every > 0 and any function v ∈ W 1,2 (Ω; [0, 1]), let us define G (v) =. Ω. (v − 1)2 |∇v| + 4 2. dx.. Our aim is to approximate F , in the sense of Γ-convergence, by a family of elliptic functionals F which are formally defined by F (u, s, σ) = (σ 2 + κ )|∇2 u|2 dx + µ |u − g|2 dx + (α − β)G (s) Ω Ω 2 + βG (σ) + ξ (s + ζ )|∇u|γ dx (3.1) Ω. for suitable nonnegative infinitesimals κ , ξ , ζ (in some cases they are allowed to vanish; see the statements below). This formula makes sense if u ∈ W 2,2 (Ω) and s, σ ∈ W 1,2 (Ω); however, in the case κ = 0, because of the coefficient σ 2 multiplying the second derivatives, the functionals F are not coercive in these spaces. In section 5 we identify a domain D(Ω) of the functionals F such that the problem.

(9) APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. (P ). 1177. min {F (u, s, σ) : (u, s, σ) ∈ D(Ω)}. has at least one solution, provided γ > 2 and κ + ζ > 0. We define X(Ω) = L2 (Ω) × L∞ (Ω; [0, 1]) × L∞ (Ω; [0, 1]) ⊃ D(Ω) and we denote by F : X(Ω) → [0, +∞] the functional defined by. F (u) if u ∈ GSBV 2 (Ω), s ≡ 1, σ ≡ 1, F(u, s, σ) = +∞ otherwise. Analogously, we denote by F : X(Ω) → [0, +∞] the functional defined by. F (u, s, σ) if (u, s, σ) ∈ D(Ω), F (u, s, σ) = +∞ otherwise. We first prove the lower inequality of Γ-convergence. Theorem 3.1. Assume that γ ≥ 2, that lim. (3.2). →0+. ξ γ−1 . = +∞,. and that either κ > 0 for small enough or ζ > 0 for small enough. Then, for every triple (u, s, σ) ∈ X(Ω) and for every family (u , s , σ ) ∈ D(Ω) converging to (u, s, σ) in [L1 (Ω)]3 as → 0+ , we have lim inf F (u , s , σ ) ≥ F(u, s, σ) . + →0. Moreover, (3.2) can be replaced by the condition ξ ≥ 0 in the case α = β. Then we prove the equicoercivity of the family (F ) under the same assumptions on γ and on the infinitesimals κ , ξ , ζ made in Theorem 3.1. Theorem 3.2. Let (u , s , σ ) ∈ D(Ω) be such that sup F (u , s , σ ) < +∞. >0. Then the family (u , s , σ ) is relatively compact in the [L1 (Ω)]3 topology as → 0+ and any limit point is of the form (u, 1, 1) with u ∈ GSBV 2 (Ω) ∩ L2 (Ω). We now consider the upper inequality of Γ-convergence. We first state our full Γ-convergence result in the special case when n = 2, γ = 2 and α = β. We recall that a domain Ω is strictly star-shaped if there exists x0 ∈ Ω such that t(Ω−x0 )+x0 ⊂⊂ Ω for any t ∈ [0, 1). Theorem 3.3. Assume that n = γ = 2, α = β, and Ω is strictly star-shaped. Assume that κ > 0 and κ = o(4 ), while ξ = ζ = 0. Then the family (F ) Γ-converges to F in the [L1 (Ω)]3 topology as → 0+ . Then from the properties of Γ-convergence and Theorem 3.2, if (¯ u , s¯ , σ ¯ ) minimizes F , then the family (¯ u , s¯ , σ ¯ ) is relatively compact in [L1 (Ω)]3 as → 0+ and any limit point corresponds to a triple (u, 1, 1) with u minimizer of F . Notice that in the case α = β, ξ = 0, the functionals F do not depend on s; hence we can write them in the much simpler form F (u, σ) = (σ 2 + κ )|∇2 u|2 dx + µ |u − g|2 dx + βG (σ) . Ω. Ω.

(10) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1178. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. Now we consider a more general situation. For every set A ⊂ Rn and every positive real number ρ, we denote by (A)ρ the open tubular neighborhood of A with radius ρ, that is, (A)ρ = {x ∈ Rn : dist(x, A) < ρ}. We define the Minkowski (n − 1)dimensional upper and lower content of the set A, respectively, by M∗ (A) = lim sup ρ→0+. meas((A)ρ ) , 2ρ. M∗ (A) = lim inf + ρ→0. meas((A)ρ ) . 2ρ. It can be shown (see [20, section 3.2.39]) that meas((A)ρ )/ρ converges to 2Hn−1 (A) as ρ → 0+ for any compact subset A of a C 1 hypersurface. In particular, by inner approximation this implies M∗ (A) ≥ Hn−1 (A) for any u ∈ BV (Ω) and any Borel set A ⊂ Su , because Hn−1 -almost all of Su can be covered by C 1 hypersurfaces. The inequality M∗ (A) ≤ Hn−1 (A), which implies lim. ρ→0+. meas((A)ρ ) = Hn−1 (A) , 2ρ. holds under very mild regularity assumptions on the set A [5]. We are able to prove the upper inequality of Γ-convergence under the assumption that u ∈ L∞ (Ω), |∇u| ∈ Lγ (Ω) and that, for the sets Su and Su ∪ S∇u , Hausdorff measure and Minkowski content coincide. Theorem 3.4. Assume that γ ≥ 2, κ = 0, ζ > 0, ξ satisfies (3.2) and ξ ζ = o(γ−1 ). Then, for every triple (u, s, σ) ∈ X(Ω) such that u ∈ L∞ (Ω), |∇u| ∈ Lγ (Ω), and M∗ (Su ) ≤ Hn−1 (Su ) ,. M∗ (Su ∪ S∇u ) ≤ Hn−1 (Su ∪ S∇u ) ,. there exist (u , s , σ ) ∈ D(Ω) converging to (u, s, σ) in [L1 (Ω)]3 as → 0+ such that (3.3). lim sup F (u , s , σ ) ≤ F(u, s, σ). →0+. Remark 3.5. The Γ-convergence result still holds if the term µ|u − g|2 in the functional F is replaced by Φ(x, u) in such a way that the functional u → Ω Φ(x, u) dx is lower semicontinuous with respect to the strong L1 (Ω) topology and continuous with respect to the strong L2 (Ω) topology (see section 7). Let Φ be a Carathéodory function on Ω × R, i.e., Φ(·, p) is measurable for any p ∈ R and Φ(x, ·) is continuous for almost every x ∈ Ω. Then a sufficient condition for Γ-convergence is the following [19]:. Φ : Ω × R → R is Carathéodory , 0 ≤ Φ(x, u) ≤ a(x) + b|u|2 , with a ∈ L1 (Ω) and b ≥ 0. 4. Basic properties of GSBV 2 functions. In this section we give some technical results concerning the one-dimensional sections of functions u ∈ GSBV (Ω). Let ν ∈ Sn−1 = {x ∈ Rn : |x| = 1} be a fixed direction. We set Πν = {x ∈ Rn : x, ν = 0}, Ωx = {t ∈ R : x + tν ∈ Ω} (x ∈ Πν ), Ων = {x ∈ Πν : Ωx = ∅}..

(11) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1179. The sets Ωx are the 1-dimensional slices of Ω indexed by x ∈ Πν , and Ων is the projection of Ω on Πν . Given u ∈ GSBV (Ω), we define for Hn−1 -a.e. x ∈ Ων the restriction ux (t) = u(x + tν). for a.e. t ∈ Ωx .. The following slicing result can be obtained from [1, Theorem 3.3] and [2, section 1]. Lemma 4.1. Let u : Ω → R be a measurable function. Then u ∈ GSBV (Ω) if and only if, for any ν ∈ Sn−1 , ux ∈ GSBV (Ωx ) for Hn−1 -a.e. x ∈ Ων and (4.1) |D(−N ∨ ux ∧ N )|(Ax ) dHn−1 < +∞ Aν. for any open set A ⊂⊂ Ω and any N ∈ N. Moreover, if u ∈ GSBV (Ω) and ν ∈ Sn−1 , then for Hn−1 -a.e. x ∈ Ων we have (a) u x (t) = ∇u(x + tν), ν for a.e. t ∈ Ωx ; (b) Sux = (Su )x . The proof of the following lemma can be found in Federer [20, section 3.2.22]. Lemma 4.2. For every countably Hn−1 -rectifiable set E ⊂ Rn there exists a Borel function νE : E → Sn−1 such that |ν, νE (x)| dHn−1 (x) = H0 (Ex ) dHn−1 (x) ∀ν ∈ Sn−1 . E. Eν. The function νE (x) is a normal unit vector to E at x in an approximate sense (see [20, section 3.2.16]). Although GSBV (Ω) is not a vector space, we can prove that the natural energy spaces for our problems do have a vector structure. Proposition 4.3. The set. n−1 |∇u| dx + H (Su ) < +∞ Y = u ∈ GSBV (Ω) : Ω. is a vector space. Proof. Let u1 , u2 ∈ Y , and ν ∈ Sn−1 be fixed. By Lemma 4.1(a), (b) and Lemma 4.2 we have |u ix | dt + H0 (Suix ) < +∞ for i = 1, 2 Ωx. for Hn−1 -a.e. x ∈ Ων , because . 0 n−1 |uix | dt + H (Suix ) dH ≤ |∇ui | dx + Hn−1 (Sui ) < +∞. Ων. Ωx. Ω. In particular, uix ∈ L∞ loc (Ωx ), and since SBVloc (Ωx ) is a vector space u1x +u2x belongs to SBVloc (Ωx ) for Hn−1 -a.e. x ∈ Ων . Since the condition (4.1) is easily verified the conclusion follows by using Lemma 4.1. Finally, we show how in GSBV 2 (Ω) second order derivatives and jump set of the derivative can be recovered as well by a slicing method. Lemma 4.4. Let u ∈ GSBV 2 (Ω) be such that |∇2 u| dx + Hn−1 (S∇u ) < +∞. Ω.

(12) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1180. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. Then, for any ν ∈ Sn−1 the function ∇u, ν belongs to GSBV (Ω) and for Hn−1 -a.e. x ∈ Ων we have (a) u x ∈ GSBV (Ωx ); (b) u. x (t) = ∇∇u, ν(x + tν), ν for a.e. t ∈ Ωx ; (c) Sux = (S∇u·ν )x . Proof. By Proposition 4.3 it follows that ∇u, ν ∈ GSBV (Ω) whenever ∇i u ∈ GSBV (Ω) for i = 1, . . . , n. By Lemma 4.1(a) it follows that u x = ∇u, νx a.e. in Ωx for Hn−1 -a.e. x ∈ Ων ; in particular, u x ∈ GSBV (Ωx ) for Hn−1 -a.e. x ∈ Ων . Then, statements (b), (c) follow by applying Lemma 4.1(a,b) to ∇u, ν. Corollary 4.5. The set. 2 2 n−1 u ∈ GSBV (Ω) : |∇ u| dx + H (Su ∪ S∇u ) < +∞ Ω. is a vector space. Proof. The proof is the same as for Proposition 4.3 using Lemma 4.4 instead of Lemma 4.1. We conclude this with an interpolation inequality in W 2,2 which provides section 2 a mild estimate of |∇u| dx with the Blake–Zisserman energy (see also [12]). Proposition 4.6. Let A, B ⊂ Rn be open sets with (A)2r ⊂⊂ B. Then (4.2). A. |∇u|2 dx ≤ 16n r−2 u2 dx + 2r2 |∇2 u|2 dx B. 2,2 ∀u ∈ Wloc (B) .. B. Proof. We prove the inequality only in the case n = 1; the general case can be achieved by a slicing argument, taking into account Lemma 4.4(b). Let x be such that the interval [x − 2r, x + 2r] ⊂ B and choose x1 ∈ [x + r, x + 2r], x2 ∈ [x − 2r, x − r] such that ru(x1 ) =. x+2r. x+r. u(s) ds,. ru(x2 ) =. x−r. x−2r. u(s) ds. and x3 ∈ [x2 , x1 ] such that u (x3 ) = [u(x1 ) − u(x2 )]/(x1 − x2 ). Then, for any y ∈ [x − 2r, x + 2r], using twice H¨ older inequality we estimate |u (y)|2 ≤ 2|u (x3 )|2 + 2. . y. x3. u. (s) ds. 2. y 4(u (x1 ) + u (x2 )). 2 + 2|x − y| |u (s)| ds 3 2 r x3 x+2r x+2r 4 ≤ 3 u2 (s) ds + 8r |u. (s)|2 ds. r x−2r x−2r. ≤. 2. 2. By integration we obtain . x+2r. x−2r. 16 |u | dy ≤ 2 r. 2. . x+2r. x−2r. 2. u dy + 32r. 2. . x+2r. x−2r. |u. |2 dy .. Covering A by a finite number of intervals of length 4r contained in B the conclusion follows..

(13) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1181. 5. The approximation framework. In this section we find a domain suitable for coercivity and lower semicontinuity of the functionals F formally defined by (3.1). We often set w = (u, s, σ) and we always assume that 0 ≤ s ≤ 1, 0 ≤ σ ≤ 1 almost everywhere. If κ = 0, we define p = 2γ/(γ + 2) and D(Ω) = (u, s, σ) ∈ X(Ω) : u, s, σ ∈ W 1,2 (Ω), σ∇u ∈ W 1,p (Ω; Rn ) , if κ > 0 we define 2,2 D(Ω) = Wloc (Ω) × W 1,2 (Ω; [0, 1]) × W 1,2 (Ω; [0, 1]) .. If u ∈ D(Ω) and κ = 0, the approximate differentiability of u and of σ∇u imply that ∇2 u exists a.e. in {σ > 0} and is given by (5.1). ∇2 u =. ∇(σ∇u) − ∇σ ⊗ ∇u . σ. We also set ∇2 u = 0 in {σ = 0}. In the following we do not need to consider the function s in the case α = β. We now prove a compactness theorem for the sublevels of F . Theorem 5.1. Assume that γ > 2, κ +ζ > 0 and let (wh ) = (uh , sh , σh ) ⊂ D(Ω) be a sequence such that sup F (wh ) < +∞. h. Then there exist a subsequence (whk ) and w = (u, s, σ) ∈ D(Ω) such that (whk ) converge in [L1 (Ω)]3 to w and (∇uhk ) converge a.e. to ∇u in {σ > 0}. Proof. From (4.2), in the case κ > 0, we have that (uh ) is bounded in W 2,2 (A) for any open set A ⊂⊂ Ω. The statement then follows from Rellich theorem. Hence, in the following we consider the more delicate case when κ = 0 and ζ > 0. From the definition of F the sequences (sh ) and (σh ) are bounded in W 1,2 (Ω). Moreover, since (|∇uh |) is bounded in Lγ (Ω) and ∇(σh ∇uh ) = σh ∇2 uh + ∇σh ⊗ ∇uh , vh = σh ∇uh are also bounded in W 1,p (Ω; Rn ). Hence, possibly extracting a further subsequence we can assume that (vhk ) is converging a.e. in Ω. It easily follows that ∇uhk = vhk /σhk converge a.e. to ∇u in {σ > 0}. In order to prove that σ∇u ∈ W 1,p (Ω; Rn ) (hence w ∈ D(Ω)), we notice that by Hölder inequality, we have |σh ∇uh | dx = 0 lim h→+∞. {σ=0}. hence (possibly extracting a subsequence) σh ∇uh converge a.e. to σ∇u in the whole of Ω. Since (σh ∇uh ) is also bounded in W 1,p (Ω; Rn ), it follows that σ∇u ∈ W 1,p (Ω; Rn ) and that σhk ∇uhk weakly converge in W 1,p (Ω; Rn ) to σ∇u. Now we prove the lower semicontinuity of F . Theorem 5.2. Assume that γ > 2, κ +ζ > 0 and let (wh ) = (uh , sh , σh ) ⊂ D(Ω) be converging in [L1 (Ω)]3 to w = (u, s, σ) ∈ D(Ω). Then lim inf F (wh ) ≥ F (w) . h→+∞.

(14) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1182. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. Proof. In this case we also consider only the more difficult case when κ = 0 and ζ > 0. It is not restrictive to assume that (F (wh )) is converging to a finite limit and, by Theorem 5.1 and its proof, we can also assume that ∇uh converge to ∇u a.e. in {σ > 0} and σh ∇uh weakly converge in W 1,p (Ω; Rn ) to σ∇u. Since the sequences (sh ) and (σh ) are bounded in W 1,2 (Ω), they weakly converge, respectively, to s, σ and therefore the terms G (s) and G (σ) are lower semicontinuous. The lower semicontinuity of Ω (s2 + ζ )|∇u|γ dx directly follows by Ioffe lower semicontinuity theorem (see [10, Theorem 4.1.1]). Finally, the identity ∇(σh ∇uh ) = σh ∇2 uh + ∇σh ⊗ ∇uh and the weak convergence of ∇(σh ∇uh ) to ∇(σ∇u) easily imply that ∇2 uh weakly converge to ∇2 u in L2 (K; Mn×n ) on any compact set K ⊂ Ω on which (σh ) uniformly converges to σ, (∇uh ) uniformly converges to ∇u, and inf K σ > 0. Then, Ioffe lower semicontinuity theorem again gives 2 2 2 σ |∇ u| dx ≤ lim inf σh2 |∇2 uh |2 dx. h→+∞. K. K. Let δ > 0; by Egorov theorem we can cover almost all of {σ ≥ δ} by an increasing sequence of compact sets on which (σh ) and (∇uh ) are uniformly converging. As a consequence, the inequality above holds with {σ≥ δ} in place of K, and letting δ ↓ 0 we obtain the lower semicontinuity of the term Ω σ 2 |∇2 u|2 dx. From the compactness and the lower semicontinuity properties of the functional F it follows that for any > 0 the problem (P ). min {F (u, s, σ) : (u, s, σ) ∈ D(Ω)}. has at least one solution, provided γ > 2 and κ + ζ > 0. Finally, if κ > 0, the problem (P ) has a solution also in the case γ = 2. 6. The lower inequality. In this section we prove the lower inequality of Γconvergence (2.4) and the equicoercivity of the family (F ). In the following it will be convenient also to consider functionals depending on the domain of integration. The following lower bound for the jump terms in the one-dimensional case has been shown by Bellettini and Coscia in [7, Theorem 3.1]. Lemma 6.1. Assume that κ , ξ , ζ are as in Theorem 3.1 and γ ≥ 2. Let I ⊂ R be a bounded open set and h → 0+ . Then, for every sequence (wh ) converging to w in [L1 (I)]3 as h → +∞ such that Fh (wh ) is bounded, we have lim inf [(α − β)Gh (sh , I) + βGh (σh , I)] ≥ (α − β)H0 (Su ∩ I) h→+∞. + βH0 ((Su ∪ Su ) ∩ I) .. (6.1). The condition (3.2) on ξ can be dropped in the case α = β. Since our functionals are slightly different from those in [7], some remarks are necessary. Indeed, the functionals in [7] are given by F (u, s, σ) = (σ 2 + κ )|∇2 u|2 dx + µ |u − g|2 dx + (α − β)G (s) Ω Ω 2 2 + βG (σ) + ξ s |∇u| dx , Ω.

(15) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1183. hence the differences with respect to ours are two: first they assume that ζ = 0 and γ = 2 and then they prove the lower bound only in the case when κ > 0 (hence uh ∈ W 2,2 (I)). The assumption that ζ = 0 is not a problem, since smaller functionals are considered, and also a general exponent γ can be considered, provided (3.2) holds. However, for technical reasons related to the proof of the Γ-limsup inequality, in particular, the difficulty in estimating the second derivatives of u = ψ u in the proof of Theorem 3.4 (this can be avoided in the case n = 1 using suitable interpolating cubic polynomials), we have preferred a different formulation of the energy in the larger class D(Ω), which still provides compactness of minimizing sequences and lower semicontinuity of the energy. Moreover, the proof of the Γ-liminf inequality of Bellettini and Coscia works, essentially with no modification, also for our more general functionals. Notice also that our full Γ-convergence result Theorem 3.3 fits exactly in the Bellettini and Coscia framework. The reason why no condition on ξ (besides ξ ≥ 0) is necessary in the case α = β is that the term ξ s2 |∇u|γ dx has been added to the energy to force σh to tend to zero at least twice (paying asymptotically at least 2β ≥ α) close to jumps of u if sh is far away from 0 (if this does not happen and (3.2) holds, then the additional term diverges; see Lemma 3.2(i) of [7]); in the case when α = β it is not necessary to force this behavior of σh , since σh is already forced by the other terms of F to tend to zero at least once (paying asymptotically at least β) close to jumps of u or of u , regardless of the values of sh . Finally, we notice that we can restate (6.1) as follows: lim inf [tFh (wh ) + (α − β)Gh (sh , I) + βGh (σh , I)] h→+∞. (6.2). ≥ (α − β)H0 (Su ∩ I) + βH0 ((Su ∪ Su ) ∩ I). ∀t > 0 .. The advantage of this new formulation is that the a priori assumption that Fh (wh ) is bounded can be dropped. 6.1. Proof of Theorem 3.1. Let (w ) ∈ D(Ω), w ∈ X(Ω), be such that w → w in [L1 (Ω)]3 as → 0+ . We assume that (6.3). +∞ > L = lim inf F (w , Ω) = lim Fh (wh , Ω) , →0. h→+∞. otherwise the result is trivial. For notational simplicity we set wh = (uh , sh , σh ) and we assume that wh converge a.e. to (u, s, σ) as h → +∞.. We also assume that (wh ) converges to w fast enough, i.e., h wh −wL1 < +∞. This assumption and Fubini theorem imply (with the notation of section 4) lim whx = wx. h→∞. a.e. in Ωx. for Hn−1 -a.e. x ∈ Ων. for any direction ν ∈ Sn−1 , and this will be useful in what follows. If either s or σ were not identically equal to 1, then by the Fatou’s lemma we would get (sh − 1)2 (σh − 1)2 L ≥ lim inf (α − β) dx + β dx ≥ +∞, h→+∞ 4h 4h {s=1} {σ=1} which contradicts the assumption that L < +∞. Therefore, we will assume that s ≡ 1 and σ ≡ 1. As before we do not need to consider the function s in the case α = β..

(16) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1184. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. The proof now follows by proving separately the following inequalities: lim inf (6.4) σh 2 |∇2 uh |2 dx ≥ |∇2 u|2 dx , h→+∞. Ω. Ω. lim inf [(α − β)Gh (sh , Ω) + βGh (σh , Ω)] ≥ (α − β)Hn−1 (Su ) h→+∞. + βHn−1 (Su ∪ S∇u ) .. (6.5). The lower semicontinuity of the term |u − g|2 dx with respect to the strong L1 (Ω) topology then completes the proof. Possibly extracting a subsequence (this is allowed, since we are assuming that Fh (wh ) is converging) we can assume that both lim inf in (6.4) and (6.5) are finite limits, denoted by L1 and L2 , respectively. t We first prove (6.4). Let ψ(t) = 0 (1 − τ ) dτ ; using (6.3) we have L+1 (1 − σh )2 h |∇σh |2 + dx ≤ |∇ψ(σh )| dx = |∇σh |(1 − σh ) dx ≤ 4 β h Ω Ω Ω for h large enough. Then, by the coarea formula (2.2), we have ψ(1) L+1 P ({ψ(σh ) > t}, Ω) dt = |∇ψ(σh )| dx ≤ β 0 Ω for h large enough. By the Fatou lemma we then get ψ(1) L+1 lim inf P ({ψ(σh ) > t}, Ω) dt ≤ lim inf |∇ψ(σh )| dx ≤ h→+∞ h→+∞ β ψ(a) Ω for any a ∈ (0, 1). Therefore there exists t0 = ψ(θ) ∈ (ψ(a), ψ(1)) for some θ ∈ (a, 1) such that lim inf P ({ψ(σh ) > t0 }, Ω) ≤ l < +∞. (6.6). h→+∞. with l = (L + 1)/[β(ψ(1) − ψ(a))]. Then, if we set Eh = {σh > θ}, by (6.6) we get P (Eh , Ω) ≤ l + 1 for infinitely many h; for notational simplicity we will assume in the following that the inequality is true for any h (in the general case a further subsequence must be extracted). By the L1 convergence of (σh ) to 1 we obtain 1 (1 − σh ) dx → 0 . meas(Ω \ Eh ) ≤ 1−θ Ω Then we define vh = uh 1Eh .. (6.7). By the locality property (2.1) we get ∇vh = 1Eh ∇uh ,. ∇2 vh = 1Eh ∇2 uh. for a.e. x ∈ Ω. Since −N ∨ vh ∧ N = 1Eh [−N ∨ uh ∧ N ]. ∀N ∈ N ,.

(17) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1185. and taking into account that Eh has finite perimeter, from [36, Chapter 4, section 6.4] it follows that vh ∈ GSBV (Ω). Analogously, −N ∨ (∇i vh ) ∧ N = 1Eh [−N ∨ (∇i uh ) ∧ N ] ∈ SBVloc (Ω) for any N ∈ N and any i = 1, . . . , n. Then vh ∈ GSBV 2 (Ω) and we have Hn−1 (Svh ∪ S∇vh ) ≤ P (Eh , Ω) ≤ l + 1. for every h ∈ N .. Then, since σh ≥ θ on Eh , the sequence (vh ) satisfies all the assumptions of the compactness Theorem 2.1, hence we can assume (again, possibly passing to a subsequence) that (vh ) converges in L1 (Ω) to some function v ∈ GSBV 2 (Ω) with ∇vh → ∇v a.e. in Ω and ∇2 vh weakly converging to ∇2 v in L2 (Ω; Mn×n ). Since (uh ) converges to u in L1 (Ω) and meas(Ω \ Eh ) → 0, we obtain that u = v ∈ GSBV 2 (Ω); moreover, by the lower semicontinuity of quadratic forms with respect to weak convergence in L2 we get L1 ≥ lim inf θ2 |∇2 uh |2 dx = lim inf θ2 |∇2 vh |2 dx ≥ θ2 |∇2 u|2 dx . h→+∞. h→+∞. Eh. Ω. Ω. By letting a ↑ 1 (hence θ → 1) we obtain (6.4). The relation (6.5) will be proved using (6.2) and a slicing argument. Let A ⊂ Ω be open and ν ∈ Sn−1 be fixed. By using the notation of section 4 we have (sh − 1)2 Gh (sh , A) ≥ h |∇sh , ν|2 + dx 4h A (shx − 1)2 = h |s hx |2 + dt dHn−1 (x) 4h Aν Ax = Gh (shx , Ax ) dHn−1 (x) . Aν. An analogous relation holds for Gh (σh , A) and, taking into account Lemma 4.4, for Fh (wh , A). Since whx converge to wx in [L1 (Ωx )]3 for Hn−1 -almost every x ∈ Ων , by using Fatou’s lemma, (6.2), Lemmas 4.1 and 4.4, and eventually Lemma 4.2, we get lim inf [tFh (wh ) + (α − β)Gh (sh , A) + βGh (σh , A)] h→+∞ ≥ lim inf [tFh (whx ) + (α − β)Gh (shx , Ax ) + βGh (σhx , Ax )] dHn−1 (x) Aν h→+∞ ≥ (α − β) H0 (Sux ∩ Ax ) dHn−1 (x) + β H0 ((Sux ∪ Sux ) ∩ Ax ) dHn−1 (x) A Aν ν 0 n−1 = (α − β) H ((Su ∩ A)x ) dH (x) + β H0 (((Su ∪ S∇u·ν ) ∩ A)x ) dHn−1 (x) Aν Aν n−1 =α |ν, νu (x)| dH (x) + β |ν, ν∇u (x)| dHn−1 (x) Su ∩A. (S∇u·ν \Su )∩A. for any t > 0, where νu (x) and ν∇u (x) are approximate unit normals to Su and S∇u , respectively. Since Fh (wh , Ω) converges to L, then by letting t ↓ 0 we obtain (6.8). lim inf [(α − β)Gh (sh , A) + βGh (σh , A)] h→+∞ n−1 ≥α |ν, νu (x)| dH (x) + β Su ∩A. (S∇u·ν \Su )∩A. |ν, ν∇u (x)| dHn−1 (x) ..

(18) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1186. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. We now apply Lemma 2.2 in the following framework: • fν (x) = α|ν, νu (x)|1Su + β|ν, ν∇u (x)|1S∇u·ν \Su , • µ = Hn−1 '(Su ∪ S∇u ) , with ν varying in a countable dense subset D of Sn−1 . Since supν∈D fν = α1Su + β1S∇u \Su (because any x ∈ S∇u belongs to S∇u·ν provided ν, ∇+ u(x)−∇− u(x) = 0), by Lemma 2.2 we have that (α − β)Hn−1 (Su ) + βHn−1 (Su ∪ S∇u ) is equal to the supremum of k α |νi , νu (x)| dHn−1 (x) + β i=1. Su ∩Ai. (S∇u·νi \Su )∩Ai. |νi , ν∇u (x)| dH. n−1. (x). among all finite families (Ai , νi ) with νi ∈ D and Ai ⊂ Ω open and pairwise disjoint. By (6.8) and the superadditivity of the lim inf operator, any of these sums is less than L2 , whence the inequality (6.5) follows (see also [5]). 6.2. Proof of Theorem 3.2. By the equiboundedness of F (w ) it follows as before that (s , σ ) → (1, 1) in [L1 (Ω)]2 as → 0+ . Reasoning as in the proof of (6.4) of Theorem 3.1 we can find a sequence h → 0+ and measurable sets Eh such that meas(Ω \ Eh ) → 0 and vh = uh 1Eh converge in older L1 (Ω) to u ∈ GSBV 2 (Ω) ∩ L2 (Ω). Since (u ) is equibounded in L2 (Ω), by H¨ inequality uh − vh L1 → 0 as h → +∞, hence uh → u in L1 (Ω). 7. The upper inequality. 7.1. Proof of Theorem 3.3. We can assume without losing generality that u ∈ GSBV 2 (Ω) ∩ L2 (Ω), |∇2 u| ∈ L2 (Ω), and σ ≡ 1. Since we are assuming that α = β we simply set s ≡ 1 for any > 0. We construct a family u converging to u in L2 (Ω), so that we can neglect the term µ |u − g|2 dx, which is continuous with respect to the strong L2 (Ω) topology, and we then assume µ = 0. Assuming that Ω is star-shaped with respect to the origin, we set Ωt = tΩ with t ∈ (0, 1) and construct a family w = (u , s , σ ) ∈ D(Ωt ) such that (as in the previous section we emphasize the dependence on the domain of integration) (7.1). lim sup F (w , Ωt ) ≤ F(w, Ω) . →0+. Then, the functions w,t (x) = w (tx) belong to D(Ω) and satisfy lim sup F (w,t , Ω) ≤ t−n F(w, Ω) , →0+. hence the desired family of the Γ-limsup inequality can be constructed by a diagonal argument by letting t ↑ 1. In order to construct the family (w ) satisfying (7.1) we follow the outline of [6], assuming first that (7.2) M∗ ((Su ∪ S∇u ) ∩ K) = Hn−1 ((Su ∪ S∇u ) ∩ K) for any K ⊂ Ω compact . We restrict our choice to the functions u and σ that, outside a tubular neighborhood of Su ∪ S∇u , with radius depending on , are, respectively, equal to u and 1..

(19) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1187. Setting S˜u = Su ∪ S∇u and τ (x) = dist(x, S˜u ), by the interpolation inequality (4.2) we obtain a constant C depending only on u such that (7.3) |∇u|2 dx ≤ Cr−2 ˜u )r Ωt \(S. for any r sufficiently small. In view of our assumption on κ , we can find an infinites1/4 that κ = o(b4 ) (for instance, b = (κ )1/2 ), and an imal b faster than such √ infinitesimal η faster than such that a = −2 ln η is infinitesimal (for instance, η = ). With this choice of infinitesimals, we then define. 0 if x ∈ (S˜u )b , σ (x) = 1 − η if x ∈ Ωt \ (S˜u )a +b . Let now y be the solution of the Cauchy problem y(t) ˙ =. 1−y , 2. y(b ) = 0,. that is, y (t) = 1 − exp [(b − t)/(2)]. We complete the definition of σ by setting σ (x) = y ◦ τ (x). if x ∈ (S˜u )a +b \ (S˜u )b .. Now we turn to the choice of u . To this aim, we build a smooth function ψ : Ω → [0, 1] such that ψ = 0 in {τ ≤ b /2}, ψ = 1 in {τ ≥ b }, and |∇ψ | = O(1/b ), |∇2 ψ | = O(1/b2 ). Taking into account that |∇τ | = 1 a.e., a function ψ with the required properties can be built as (χ ◦ τ ) ∗ ρ, where ρ is a convolution kernel with diameter b /3 and χ(s) = [0 ∨ (6s/b − 4) ∧ 1]. The assumptions on u and the interpolation inequality (4.2) yield 2,2 u ∈ Wloc (Ω \ S˜u ) ,. so that, if we set u = uψ , we have u ∈ W 2,2 (Ωt ). With these choices, we get (7.4) (σ2 + κ )|∇2 u |2 dx F (w , Ωt ) = Ωt. +βG (σ , Ωt ∩ ((S˜u )a +b \ (S˜u )b )) meas(Ωt ∩ (S˜u )b ) +β 4 η2 +β meas(Ωt \ (S˜u )a +b ) . 4. (7.5) (7.6) (7.7). Since u ≡ u on {σ > 0} (because ψ ≡ 1 on {τ ≥ b } ⊃ {σ > 0}), the upper limit of the term in (7.4) does not exceed 2 2 |∇ u| dx + lim sup κ |∇2 u |2 dx . Ω. →0+. Taking into account (7.3), the identity. Ωt ∩{b /2≤τ ≤b }.

(20) 1188. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. ∇2 u = ψ ∇2 u + 2∇ψ ⊗ ∇u + u∇2 ψ , and our choice of b we obtain that the lim sup above is zero. Concerning the term in (7.5), from the proof of Theorem 3.1 of [6] it follows that its upper limit does not exceed βM∗ (S˜u ∩ Ωt ); by (7.2) we obtain that the upper limit is less than βHn−1 (S˜u ). The term in (7.6) is infinitesimal because the Minkowski content is finite and b = o(), and similarly the term in (7.7) is infinitesimal because η2 = o(). This proves, under the additional assumption (7.2), the existence of a family (w ) satisfying (7.1). The assumption can be removed as follows. Consider for any λ > 0 the penalized problem. . 2 2 min |∇ v| + λ|v − u|2 dx + βHn−1 (Sv ∪ S∇v ) : v ∈ GSBV 2 (Ω) ∩ L2 (Ω) , Ω. and let uλ be a minimizer (see [12]). Notice that F (uλ ) ≤ F (u) < +∞, hence uλ → u as λ → +∞. Then, it has been proved in [14] that any function uλ fulfills (7.2), and therefore a family (w ) satisfying (7.1) for (u, 1, 1) can be obtained from those already constructed for (uλ , 1, 1) by a diagonal argument. 7.2. Proof of Theorem 3.4. Since the proof is similar to that of Theorem 3.3 we sketch only the relevant differences. The function σ is defined in the same way and s is constructed analogously in a tubular neighborhood of Su . Let τ1 (x) = dist(x, Su ). In order to construct u we fix some smooth function ψ such that ψ = 0 in {τ1 ≤ b /2}, ψ = 1 in {τ1 ≥ b }, and |∇ψ | = O(1/b ). The assumptions on u yield u ∈ W 1,γ (Ω \ S u ), so that setting u = uψ we have u ∈ W 1,γ (Ω) and σ ∇u ∈ W 1,p (Ω; Rn ). The cut-off function ψ is built only in the tubular neighborhood of Su , otherwise the term ξ (s2 + ζ )|∇u |γ dx cannot be controlled in the neighborhood of S∇u \ Su . Then we must set κ = 0 otherwise F (w ) is not finite. With these choices the estimates proceed in the same way as in the proof of Theorem 3.3 taking into account that the upper limit of the term ξ (s2 +ζ )|∇u |γ dx does not exceed ξ |∇u|γ dx + lim sup ξ ζ |∇u |γ dx . Ω. →0+. Ω∩{b /2≤τ1 ≤b }. In view of the assumption on ξ ζ , we can find an infinitesimal b faster than such 1 that ξ ζ = o(bγ−1 ), for instance, b = ((ξ ζ ) γ−1 )1/2 , so that the lim sup above is zero. 8. An application to the computation of depth from stereo images. The Γ-convergent approximation has been experimented on the problem of computation of depth from a pair of stereo images for the purpose of illustration. In the following Ω denotes the open set (0, 1) × (0, 1) of R2 , and x = (x1 , x2 ). In the case of parallel camera geometry [22] we choose the expression of the function Φ(x, u) used in [23, 24, 25]: 2. Φ(x, u) = µ [L(x1 , x2 ) − R(x1 + u, x2 )] , where u is the disparity function, µ > 0 is a parameter, and R, L are bounded continuous functions corresponding to the right and left image intensities. Depth is.

(21) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. 1189. inversely proportional to disparity. The Γ-convergence theorem may be applied if the functions R and L satisfy the conditions of the Remark 3.5, which can be fulfilled, for instance, by means of a convolution of the image intensities with a smooth kernel having a suitably small diameter. For the purpose of illustration we set γ = 2. A simple discretization method, commonly used for computer vision problems [35], may be applied to the functionals F in a straightforward way. Discrete versions of u, s, and σ are defined on a square lattice of coordinates (ih, jh), where h = h 1/(N − 1), 0 ≤ i ≤ N − 1, 0 ≤ j ≤ N − 1. We denote by uhi,j , shi,j , and σi,j , an approximation of u(ih, jh), s(ih, jh), and σ(ih, jh), respectively. We denote by 2 uh , sh , σ h ∈ RN the vectors of the discrete variables. Then we set κ = 0, ζ > 0, and we discretize F1 (u, s, σ) = σ 2 |∇2 u|2 dx + ξ (s2 + ζ )|∇u|2 dx Ω. by 1 F,h (uh , sh , σ h ). =. . h 2 (σi,j ). i,j. Ω. 1 h (ui+1,j − 2uhi,j + uhi−1,j )2 h2. + 2(uhi+1,j+1 − uhi,j+1 − uhi+1,j + uhi,j )2 + (uhi,j+1 − 2uhi,j + uhi,j−1 )2 +. (8.1). ξ ((shi,j )2. h h 2 h h 2 . + ζ ) (ui+1,j − ui,j ) + (ui,j+1 − ui,j ). We set F2 (s, σ) = (α − β)G (s) + βG (σ) , and we discretize G (s) by. h2 h h h h 2 h h 2 2 (8.2) G,h (s ) = (si+1,j − si,j ) + (si,j+1 − si,j ) + (si,j − 1) , 4 i,j and analogously for G (σ). Then we set 2 F 3 (u) = µ [L(x1 , x2 ) − R(x1 + u(x1 , x2 ), x2 )] dx , Ω. which is discretized by (8.3). Fh3 (uh ) = µ. i,j. h h2 Lhi,j − Ri+u h. ,j i,j. 2. ,. h where Ri,j , Lhi,j denote an approximation of R(ih, jh), L(ih, jh) and, since uhi,j is generally not an integer, the discretization of R is computed by means of a linear interpolation. We set 1 2 (uh , sh , σ h ) + F,h (sh , σ h ) + Fh3 (uh ) . F,h (uh , sh , σ h ) = F,h. In order to recover a stable solution, the grid must resolve the width of the transition region of the functions s and σ. Then the discretization step should be.

(22) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1190. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. at least h = o() as it has been shown in [8] for the discretization of the Ambrosio and Tortorelli approximating functionals. A global solution of the discrete nonconvex variational problem could be computed by means of a stochastic optimization method. However, we use a faster deterministic continuation procedure in which α and β are considered as continuation variables [35]. The functional F1 +F2 becomes increasingly convex for larger values of these variables. Then a solution of the system of equations ∇F,h (uh , sh , σ h ) = 0 is computed by using a nonlinear Gauss–Seidel iterative method, with α and β initially set to high values, then gradually lowered. The continuation procedure yields experimental good, although not globally optimal, solutions. The parameters α and β are lowered according to the rule (8.4). α(k) = α0 (c)k ,. β (k) = β0 (c)k ,. where α0 , β0 are the initial values, c < 1 is a real positive number, and each step k of the continuation procedure consists of 32 nonlinear Gauss–Seidel iterations. The Γ-convergence theorem and the continuation algorithm have been experimented on synthetic stereo pairs of images corresponding to simple patterns. The images are discretized with N =256. The brightness patterns of all the surfaces represented in the synthetic images are linear combinations of spatially orthogonal sinusoids. The spatial frequency of the sinusoids is chosen to give a reasonably strong brightness gradient such as that usually required for binocular stereo matching (see also March [23, 24, 25]). The range of brightness values for L, R is [0,255]. Depth has to be recovered from the local geometrical distortion of the brightness pattern in the left image relative to the one in the right image. The periodicity of the brightness pattern causes further difficulties to the problem of recovering disparity because of the presence of many ambiguous corresponding points in the two images. The algorithm was started with an initial estimate of the disparity function u equal to a constant value, and setting the functions σ, s equal to 1 everywhere. The values of the parameters in the functional were chosen on the basis of the results of a number of experiments. Figures 1(a) and 1(b) show the two images L and R of a stereo pair representing an object shaped as a revolution surface and portrayed against a plane background. The value of disparity ranges from 14 to 32 pixels (14h ≤ u ≤ 32h). The stereo disparity u in the images of Figures 1(a) and 1(b) is discontinuous along the occluding boundary between the curved surface and the plane background. The function u has no creases in this example. √ We set µ = 26 and c = 0.8 in (8.4). The continuation procedure is iterated for 43 steps (1376 total Gauss–Seidel iterations) and the final values of the parameters are α = 37, β = 32.7. Figure 2(a) shows the function σ computed with = 2 · 10−2 , and Figure 2(b) shows the same function computed with = 1.3 · 10−2 . Figure 3(a) shows the function σ computed with = 6.5 · 10−3 : in this case σ reaches values of order 10−5 along the discontinuity set of u. In the figures representing the functions σ and s by means of grey values, white corresponds to 0 and black corresponds to 1. The figures show the convergence of the functions σ towards the discontinuity set of the disparity u as decreases, thus illustrating the behavior of Γ-convergence in this specific example. Figure 3(b) shows the function s computed with = 6.5 · 10−3 . Because of the presence of the factor ξ converging to zero, the values of the functions s might approach zero more slowly than the functions σ as tends to zero..

(23) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. (a) L image of a synthetic stereo pair (only jumps).. 1191. (b) R image of a synthetic stereo pair (only jumps).. Fig. 1.. (a) The function σ computed with = 2.0 · 10−2 . Fig. 2.. (b) The function σ computed with = 1.3 · 10−2 ..

(24) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1192. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. (a) The function σ computed with = 6.5 · 10−3 .. (b) The function s computed with = 6.5 · 10−3 .. Fig. 3.. Fig. 4. Surfaces with jumps recovered from the stereo pair 1(a), (b)..

(25) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. (a) L image of a synthetic stereo pair (only creases).. 1193. (b) R image of a synthetic stereo pair (only creases).. Fig. 5.. (a) The function σ computed with = 8.0 · 10−3 . Fig. 6.. (b) The function σ computed with = 2.0 · 10−3 ..

(26) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1194. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. Fig. 7. Surface with creases recovered from the stereo pair 5(a), (b).. (a) L image of a synthetic stereo pair (jumps+creases). Fig. 8.. (b) R image of a synthetic stereo pair (jumps+creases)..

(27) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. APPROXIMATION OF A FREE DISCONTINUITY PROBLEM. (a) The function σ computed with = 2.0 · 10−3 .. 1195. (b) The function s computed with = 2.0 · 10−3 .. Fig. 9.. Fig. 10. Surface with jumps and creases recovered from the stereo pair 8(a), (b)..

(28) Downloaded 09/17/18 to 192.167.204.51. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1196. LUIGI AMBROSIO, LORIS FAINA, AND RICCARDO MARCH. For instance, in the present example s reaches the value of 6 · 10−2 , but takes values between 0.3 and 0.7 along some portions of the discontinuity contour. Finally, Figure 4 shows the surfaces corresponding to the disparity map recovered from the stereo pair. The discontinuity set is correctly reconstructed along the occluding contour. Figures 5(a) and 5(b) show the images L, R of a stereo pair representing a truncated pyramid laid upon a plane background. Disparity ranges from 12 to 24 pixels (12h ≤ u ≤ 24h) and, in this example, the function u has creases and no jumps. √ We set µ = 47 and c = 0.76. The continuation procedure consists of 45 steps (1440 total Gauss–Seidel iterations) and the final values of the parameters are α = 14.6, β = 7.3. Figure 6(a) shows the function σ computed with = 8 · 10−3 . Figure 6(b) shows σ computed with = 2 · 10−3 : in this case σ reaches values of order either 10−3 or 10−2 along the set of creases of u. The figures show the capability of Γ-convergence in the localization of the creases. Figure 7 shows the surface recovered from the stereo pair with the creases correctly reconstructed. Figures 8(a) and 8(b) show the last stereo pair used in the computer experiments which is obtained from the previous one by introducing a jump between the truncated pyramid and the plane background. Disparity ranges from 8 to 24 pixels (8h ≤ u ≤ 24h). In this example the function u has both creases and jumps. √ We set µ = 47 and c = 0.76. The continuation procedure consists of 43 steps (1376 Gauss–Seidel iterations) and the final values of the parameters are α = 14.3, β = 12.6. Figures 9(a) and 9(b) show, respectively, the functions σ and s computed with = 2 · 10−3 . The function σ reaches values of order 10−5 along the jumps, and of order 10−2 along the creases, while s is about 0.3 along the jumps. Finally, Figure 10 shows the surfaces corresponding to the disparity map recovered from the stereo pair. Both discontinuities and gradient discontinuities are reconstructed. Acknowledgments. The authors wish to thank an anonymous referee for the careful review of the paper which improved the presentation. The third author wishes to thank Dr. Mario Rosati for many discussions. REFERENCES [1] L. Ambrosio, A compactness theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital., 3–B (1989), pp. 857–881. [2] L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal., 111 (1990), pp. 291–322. [3] L. Ambrosio, The space SBV (Ω) and free discontinuity problems, in Variational and Free Boundary Problems, A. Friedman and J. Spruck eds., IMA Vol. Math. Appl. 53, Springer, New York, 1993, pp. 29–45. [4] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University Press, New York, 2000. [5] L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math., 43 (1990), pp. 999–1036. [6] L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Un. Mat. Ital. B (7), 6 (1992), pp. 105–123. [7] G. Bellettini and A. Coscia, Approximation of a functional depending on jumps and corners, Boll. Un. Mat. Ital. B (7), 8 (1994), pp. 151–181. [8] G. Bellettini and A. Coscia, Discrete approximation of a free discontinuity problem, Numer. Funct. Anal. Optim., 15 (1994), pp. 201–224. [9] A. Blake and A. Zisserman, Visual Reconstruction, MIT Press, Cambridge, MA, 1987. [10] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes Math. Ser. 207, Longman, Harlow, UK, 1989. [11] A. P. Calderon and A. Zygmund, On the differentiability of functions which are of bounded.

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